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G. J. ZALMAI

Received 28 December 2003 and in revised form 3 October 2004

We construct a number of semiparametric duality models and establish appropriate du- ality results under various generalized (,b,φ,ρ,θ)-univexity assumptions for a multiob- jective fractional subset programming problem.

1. Introduction

In this paper, we will present a number of semiparametric duality results under various generalized (,b,φ,ρ,θ)-univexity hypotheses for the following multiobjective fractional subset programming problem:

(P)

) subject to Hj(S) 0, j ∈ q, S∈An, (1.1)

whereAn is the n-fold product of the σ-algebraA of subsets of a given set X ,Fi,Gi, i∈ p ≡ {1,2, . . . , p}, and Hj , j ∈ q, are real-valued functions defined on An, and for each i ∈ p, Gi(S) > 0 for all S∈An such that Hj(S) 0, j ∈ q.

This paper is essentially a continuation of the investigation that was initiated in the companion paper [6] where some information about multiobjective fractional program- ming problems involving point-functions as well as n-set functions was presented, a fairly comprehensive list of references for multiobjective fractional subset programming prob- lems was provided, a brief overview of the available results pertaining to multiobjective fractional subset programming problems was given, and numerous sets of semiparamet- ric sufficient efficiency conditions under various generalized (,b,φ,ρ,θ)-univexity as- sumptions were established. These and some other related material that were discussed in [6] will not be repeated in the present paper. Making use of the semiparametric sufficient efficiency criteria developed in [6] in conjunction with a certain necessary efficiency re- sult that will be recalled in the next section, here we will construct several semiparametric duality models for (P) with varying degrees of generality and, in each case, prove appro- priate weak, strong, and strict converse duality theorems under a number of generalized (,b,φ,ρ,θ)-univexity conditions.

Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:7 (2005) 1109–1133 DOI: 10.1155/IJMMS.2005.1109

1110 Multiobjective fractional subset programming

The rest of this paper is organized as follows. In Section 3 we consider a simple dual problem and prove weak, strong, and strict converse duality theorems. In Section 4 we formulate another dual problem with a relatively more flexible structure that allows for a greater variety of generalized (,b,φ,ρ,θ)-univexity conditions under which duality can be established. In Sections 5 and 6 we state and discuss two general duality models which are, in fact, two families of dual problems for (P), whose members can easily be identified by appropriate choices of certain sets and functions.

Evidently, all of these duality results are also applicable, when appropriately special- ized, to the following three classes of problems with multiple, fractional, and conventional objective functions, which are particular cases of (P):

(P1)

F1(S), (1.4)

where F (assumed to be nonempty) is the feasible set of (P), that is,

F= {S∈An :Hj(S) 0, j ∈ q}. (1.5)

Since in most cases the duality results established for (P) can easily be modified and restated for each one of the above problems, we will not explicitly state these results.

2. Preliminaries

In this section, we gather, for convenience of reference, a few basic definitions and auxil- iary results which will be used frequently throughout the sequel.

Let (X ,A,µ) be a finite atomless measure space with L1(X ,A,µ) separable, and let d be the pseudometric on An defined by

d(R,S)= [ n∑ i=1

)∈An, (2.1)

where denotes symmetric difference; thus (An,d) is a pseudometric space. For h ∈ L1(X ,A,µ) and T ∈ A with characteristic function χT ∈ L∞(X ,A,µ), the integral

∫ T hdµ

will be denoted by h,χT. We next define the notion of differentiability for n-set functions. It was originally in-

troduced by Morris [3] for a set function, and subsequently extended by Corley [1] for n-set functions.

G. J. Zalmai 1111

Definition 2.1. A function F : A → R is said to be differentiable at S∗ if there exists DF(S∗)∈ L1(X ,A,µ), called the derivative of F at S∗, such that for each S∈A,

F(S)= F(S∗)+ ⟨ DF

( S∗ ) ,χS− χS∗

where VF(S,S∗) is o(d(S,S∗)), that is, limd(S,S∗)→0VF(S,S∗)/d(S,S∗)= 0.

Definition 2.2. A function G : An → R is said to have a partial derivative at S∗ = (S∗1 , . . . , S∗n )∈An with respect to its ith argument if the function F(Si)=G(S∗1 , . . . ,S∗i−1,Si,S∗i+1, . . . , S∗n ) has derivativeDF(S∗i ), i∈ n; in that case, the ith partial derivative ofG at S∗ is defined to be DiG(S∗)=DF(S∗i ), i∈ n.

Definition 2.3. A function G : An → R is said to be differentiable at S∗ if all the partial derivatives DiG(S∗), i∈ n, exist and

G(S)=G(S∗)+ n∑ i=1

⟨ DiG

⟩ +WG

( S,S∗

) , (2.3)

where WG(S,S∗) is o(d(S,S∗)) for all S∈An. We next recall the definitions of the generalized (,b,φ,ρ,θ)-univex n-set functions

which will be used in the statements of our duality theorems. For more information about these and a number of other related classes of n-set functions, the reader is referred to [6]. We begin by defining a sublinear function which is an integral part of all the subsequent definitions.

Definition 2.4. A function : Rn → R is said to be sublinear (superlinear) if (x + y) ()(x) + (y) for all x, y ∈Rn, and (ax)= a(x) for all x ∈Rn and a∈R+ ≡ [0,∞).

Let S,S∗ ∈An, and assume that the function F :An→R is differentiable at S∗.

Definition 2.5. The function F is said to be (strictly) (,b,φ,ρ,θ)-univex at S∗ if there exist a sublinear function (S,S∗;·) : Ln1(X ,A,µ)→ R, a function b : An ×An → R with positive values, a function θ : An ×An → An ×An such that S = S∗ ⇒ θ(S,S∗) = (0,0), a function φ :R→R, and a real number ρ such that for each S∈An,

φ ( F(S)−F(S∗))(>)

+ ρd2(θ(S,S∗ )) . (2.4)

Definition 2.6. The function F is said to be (strictly) (,b,φ,ρ,θ)-pseudounivex at S∗ if there exist a sublinear function (S,S∗;·) : Ln1(X ,A,µ) → R, a function b : An ×An → R with positive values, a function θ : An ×An → An ×An such that S = S∗ ⇒ θ(S,S∗) = (0,0), a function φ :R→R, and a real number ρ such that for each S∈An (S = S∗),

( S,S∗;b

Definition 2.7. The function F is said to be (prestrictly) (,b,φ,ρ,θ)-quasiunivex at S∗

if there exist a sublinear function (S,S∗;·) : Ln1(X ,A,µ)→ R, a function b : An ×An → R with positive values, a function θ : An ×An → An ×An such that S = S∗ ⇒ θ(S,S∗) = (0,0), a function φ :R→R, and a real number ρ such that for each S∈An,

φ ( F(S)−F(S∗))(<) 0=⇒

1112 Multiobjective fractional subset programming

From the above definitions it is clear that if F is (,b,φ,ρ,θ)-univex at S∗, then it is both (,b,φ,ρ,θ)-pseudounivex and (,b,φ,ρ,θ)-quasiunivex at S∗, if F is (,b,φ,ρ,θ)- quasiunivex at S∗, then it is prestrictly (,b,φ,ρ,θ)-quasiunivex at S∗, and if F is strictly (,b,φ,ρ,θ)-pseudounivex at S∗, then it is (,b,φ,ρ,θ)-quasiunivex at S∗.

In the proofs of the duality theorems, sometimes it may be more convenient to use certain alternative but equivalent forms of the above definitions. These are obtained by considering the contrapositive statements. For example, (,b,φ,ρ,θ)-quasiunivexity can be defined in the following equivalent way: F is said to be (,b,φ,ρ,θ)-quasiunivex at S∗

if for each S∈An,

( S,S∗;b

))=⇒ φ ( F(S)−F(S∗)) > 0. (2.7)

Needless to say, the new classes of generalized convex n-set functions specified in Def- initions 2.5, 2.6, and 2.7 contain a variety of special cases; in particular, they subsume all the previously defined types of generalized n-set functions. This can easily be seen by appropriate choices of , b, φ, ρ, and θ.

In the sequel we will also need a consistent notation for vector inequalities. For all a,b ∈ Rm, the following order notation will be used: a b if and only if ai bi for all i∈m; a b if and only if ai bi for all i∈m, but a = b; a > b if and only if ai > bi for all i∈m; a b is the negation of a b.

Throughout the sequel we will deal exclusively with the efficient solutions of (P). An x∗ ∈ is said to be an efficient solution of (P) if there is no other x ∈ such that (x) (x∗), where is the objective function of (P).

Next, we recall a set of parametric necessary efficiency conditions for (P).

Theorem 2.8 [5]. Assume that Fi,Gi, i ∈ p, and Hj , j ∈ q, are differentiable at S∗ ∈ An,

and that for each i∈ p, there exist Si ∈An such that

Hj ( S∗ )

⟨ DkF

⟩ < 0. (2.9)

If S∗ is an efficient solution of (P) and λ∗i = (S∗), i ∈ p, then there exist u∗ ∈ U = {u ∈ Rp : u > 0,

∑p i=1ui = 1} and v∗ ∈Rq

+ such that

⟩ 0, ∀S∈An,

G. J. Zalmai 1113

The above theorem contains two sets of parameters u∗i and λ∗i , i ∈ p, which were in- troduced as a consequence of our indirect approach in [5] requiring two intermediate auxiliary problems. It is possible to eliminate one of these two sets of parameters and thus obtain a semiparametric version of Theorem 2.8. Indeed, this can be accomplished by simply replacing λ∗i by Fi(S∗)/Gi(S∗), i ∈ p, and redefining u∗ and v∗. For future reference, we state this in the next theorem.

Theorem 2.9. Assume that Fi,Gi, i ∈ p, and Hj , j ∈ q, are differentiable at S∗ ∈ An, and

that for each i∈ p, there exist Si ∈An such that

Hj ( S∗ )

n∑ k=1

⟨ Gi ( S∗ ) DkF

( S∗ )−Fi(S∗)DkG

⟩ < 0. (2.12)

If S∗ is an efficient solution of (P), then there exist u∗ ∈U and v∗ ∈Rq + such that

n∑ k=1

⟨ p∑ i=1

( S∗ )−Fi(S∗)DkGi

⟩ 0, ∀S∈An,

v∗j Hj ( S∗ )= 0, j ∈ q.

For simplicity, we will henceforth refer to an efficient solution S∗ of (P) satisfying (2.11) and (2.12) for some Si, i∈ p, as a normal efficient solution.

The form and contents of the necessary efficiency conditions given in Theorem 2.9 in conjunction with the sufficient efficiency results established in [6] provide clear guide- lines for constructing various types of semiparametric duality models for (P).

3. Duality model I

In this section, we discuss a duality model for (P) with a somewhat restricted constraint structure that allows only certain types of generalized (,b,φ,ρ,θ)-univexity conditions for establishing duality. More general duality models will be presented in subsequent sec- tions.

In the remainder of this paper, we assume that the functions Fi,Gi, i∈ p, andHj , j ∈ q, are differentiable onAn and that Fi(T) 0 andGi(T) > 0 for each i∈ p and for all T such that (T ,u,v) is a feasible solution of the dual problem under consideration.

1114 Multiobjective fractional subset programming

Consider the following problem: (DI)

Minimize ( F1(T) G1(T)

T ∈An, u∈U , v ∈Rq +, (3.4)

where (S,T ;·) : Ln1(X ,A,µ)→R is a sublinear function. The following two theorems show that (DI) is a dual problem for (P).

Theorem 3.1 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DI), respectively, and assume that any one of the following three sets of hypotheses is satis- fied:

(a) (i) for each i ∈ p, Fi is (,b, φ, ρi,θ)-univex at T , and −Gi is (,b, φ, ρi,θ)-univex

at T , φ is superlinear, and φ(a) 0⇒ a 0; (ii) for each j ∈ q, Hj is (,b, φ, ρ,θ)-univex at T , φ is increasing, and φ(0)= 0;

(iii) ρ∗ + ∑

j∈J+ vj ρ j 0, where ρ∗ =∑p i=1ui[Gi(T)ρi +Fi(T)ρi];

(b) (i) for each i ∈ p, Fi is (,b, φ, ρi,θ)-univex at T , and −Gi is (, b, φ, ρi,θ)-univex

at T , φ is superlinear, and φ(a) 0⇒ a 0; (ii) the function T →∑q

j=1 vjHj(T) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing,

and φ(0)= 0; (iii) ρ∗ + ρ 0;

(c) (i) the Lagrangian-type function

T −→ p∑ i=1

vjHj(T), (3.5)

where S is fixed in An, is (,b, φ,0,θ)-pseudounivex at T and φ(a) 0⇒ a 0. Then (S) ξ(T ,u,v), where ξ = (ξ1, . . . ,ξp) is the objective function of (DI).

Proof. (a) From (i) and (ii) it follows that

φ ( Fi(S)−Fi(T)

G. J. Zalmai 1115

Multiplying (3.6) by uiGi(T) and (3.7) by uiFi(T), i∈ p, adding the resulting inequalities,

and then using the superlinearity of φ and sublinearity of (S,T ;·), we obtain

φ

φ

(3.10)

Since v 0, S∈ F, and (3.3) holds, it is clear that

q∑ j=1

vj [ Hj(S)−Hj(T)

] 0, (3.11)

which implies, in view of the properties of φ, that the left-hand side of (3.10) is less than or equal to zero, that is,

0

vjDHj(T)

) 0.

(3.13)

Now adding (3.9) and (3.12), and then using (3.13) and (iii), we obtain

φ

p∑ i=1

ui [ Gi(T)Fi(S)−Fi(T)Gi(S)

Since u > 0, (3.15) implies that( G1(T)F1(S)−F1(T)G1(S), . . . ,Gp(T)Fp(S)−Fp(T)Gp(S)

) (0, . . . ,0), (3.16)

(S)= ( F1(S) G1(S)

) = ξ(T ,u,v). (3.17)

(b) Since for each j ∈ q, vjHj(S) 0, it follows from (3.3) that

q∑ j=1

vjHj(T), (3.18)

φ

vjDHj(T)

) −ρd2(θ(S,T)

) . (3.20)

Now combining (3.9), (3.13), and (3.20), and using (iii), we obtain (3.15). Therefore, the rest of the proof is identical to that of part (a).

(c) From the (,b, φ,0,θ)-pseudounivexity assumption and (3.2) it follows that

φ

In view of the properties of φ, this inequality becomes

p∑ i=1

ui [ Gi(T)Fi(S)−Fi(T)Gi(S)

vjHj(T) 0, (3.22)

which because of (3.3), primal feasibility of S, and nonnegativity of v, reduces to (3.15), and so the rest of the proof is identical to that of part (a).

Theorem 3.2 (strong duality). Let S∗ be a regular efficient solution of (P), let (S,S∗; DF(S∗)) =∑n

k=1DkF(S∗),χSk − χS∗k for any differentiable function F : An → R and S ∈ An, and assume that any one of the three sets of hypotheses specified in Theorem 3.1 holds for all feasible solutions of (DI). Then there exist u∗ ∈U and v∗ ∈Rq

+ such that (S∗,u∗,v∗) is an efficient solution of (DI) and (S∗)= ξ(S∗,u∗,v∗).

G. J. Zalmai 1117

Proof. By Theorem 2.9, there exist u∗ ∈U and v∗ ∈Rq + such that (S∗,u∗,v∗) is a feasible

solution of (DI). If it were not an efficient solution, then there would exist a feasible solution (T , u, v) such that ξ(T , u, v) ξ(S∗,u∗,v∗)= (S∗), which contradicts the weak duality relation established in Theorem 5.1. Therefore, (S∗,u∗,v∗) is an efficient solution of (DI).

We also have the following converse duality result for (P) and (DI).

Theorem 3.3 (strict converse duality). Let S∗ and (S,S∗;·) be as in Theorem 3.2, let (S, u, v) be a feasible solution of (DI) such that

p∑ i=1

ui [ Gi(S)Fi(S∗)−Fi(S)Gi(S∗)

] 0. (3.23)

Furthermore, assume that any one of the following three sets of hypotheses is satisfied: (a) the assumptions specified in part (a) of Theorem 3.1 are satisfied for the feasible so-

lution (S, u, v) of (DI); Fi is strictly (,b, φ, ρi,θ)-univex at S for at least one index i ∈ p with the corresponding component ui of u positive, and φ(a) > 0⇒ a > 0, or

−Gi is strictly (,b, φ, ρi,θ)-univex at S for at least one index i ∈ p with ui posi-

tive, and φ(a) > 0⇒ a > 0, or Hj is strictly (,b, φ, ρ j ,θ)-univex at S for at least one

index j ∈ q with v j positive, and φ(a) > 0⇒ a > 0, or ∑p

i=1 ui[Gi(S)ρi + Fi(S)ρi] +∑q j=1 v j ρ j > 0;

(b) the assumptions specified in part (b) of Theorem 3.1 are satisfied for the feasible solu- tion (S, u, v) of (DI), Fi and φ or−Gi and φ satisfy the requirements described in part (a), or the function R→∑q

j=1 v jHj(R) is strictly (,b, φ, ρ,θ)-pseudounivex at S, or∑p i=1 ui[Gi(S)ρi +Fi(S)ρi] + ρ > 0;

(c) the assumptions specified in part (c) of Theorem 3.1 are satisfied for the feasible solu- tion (S, u, v) of (DI), and the function

R−→ p∑ i=1

q∑ j=1

v jHj(R) (3.24)

is strictly (,b, φ,0,θ)-pseudounivex at S, and φ(a) > 0⇒ a > 0. Then S= S∗, that is, S is an efficient solution of (P).

Proof. (a) Suppose to the contrary that S = S∗. Proceeding as in the proof of part (a) of Theorem 5.1, we arrive at the strict inequality

p∑ i=1

ui [ Gi(S)Fi(S∗)−Fi(S)Gi

]= 0, (3.25)

in contradiction to (3.23). Hence we conclude that S= S∗. (b) and (c) The proofs are similar to that of part (a).

1118 Multiobjective fractional subset programming

4. Duality model II

In this section, we consider a slightly different version of (DI) that allows for a greater variety of generalized (,b,φ,ρ,θ)-univexity conditions under which duality can be es- tablished. This duality model has the form

(DII)

T ∈An, u∈U0, v ∈Rq +, (4.4)

where (S,T ;·) : Ln1(X ,A,µ) → R is a sublinear function, and U0 = {u ∈ Rq : u 0,∑p i=1ui = 1}. We next show that (DII) is a dual problem for (P) by establishing weak and strong

duality theorems. As demonstrated below, this can be accomplished under numerous sets of generalized (,b,φ,ρ,θ)-univexity conditions. Here we use the functions fi(·,S), i∈ p, f (·,S,u), and h(·,v) :An→R, which are defined, for fixed S, u, and v, as follows:

fi(T ,S,u)=Gi(S)Fi(T)−Fi(S)Gi(T), i∈ p,

f (T ,S,u)= p∑ i=1

ui [ Gi(S)Fi(T)−Fi(S)Gi(T)

vjHj(T).

(4.5)

For given u∗ ∈ U0 and v∗ ∈ Rq +, let I+(u∗)= {i∈ p : u∗i > 0} and J+(v∗)= { j ∈ q : v∗j >

0}. Theorem 4.1 (weak duality). Let S and (T ,u,v), with u > 0, be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following six sets of hypotheses is satisfied:

(a) (i) f (·,T ,u) is (,b, φ, ρ,θ)-pseudounivex at T , and φ(a) 0⇒ a 0; (ii) for each j ∈ J+ ≡ J+(v), Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing,

and φ j(0)= 0; (iii) ρ+

∑ j∈J+ vj ρ j 0;

(b) (i) f (·,T ,u) is (,b, φ, ρ,θ)-pseudounivex at T , and φ(a) 0⇒ a 0; (ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0;

(iii) ρ+ ρ 0;

G. J. Zalmai 1119

(c) (i) f (·,T ,u) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each j ∈ J+, Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing, and

φ j(0)= 0; (iii) ρ+

∑ j∈J+ vj ρ j > 0;

(d) (i) f (·,T ,u) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0;

(iii) ρ+ ρ > 0;

(e) (i) f (·,T ,u) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each j ∈ J+, Hj is strictly (,b, φ j , ρ j ,θ)-pseudounivex at T , φ j is increasing,

and φ j(0)= 0; (iii) ρ+

∑ j∈J+ vj ρ j 0;

(f) (i) f (·,T ,u) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0;

(ii) h(·,v) is strictly (, b, φ, ρ,θ)-pseudounivex at T , φ is increasing, and φ(0)= 0; (iii) ρ+ ρ 0.

Then (S) ψ(T ,u,v), where ψ = (ψ1, . . . ,ψp) is the objective function of (DII).

Proof. (a) From the primal feasibility of S and (4.3) it is clear that for each j ∈ J+,Hj(S) Hj(T) and so using the properties of φ j , we obtain φ j(Hj(S)−Hj(T)) 0, which by virtue of (ii) implies that for each j ∈ J+,

( S,T ;b(S,T)DHj(T)

) −ρ jd2(θ(S,T)

) . (4.6)

(4.8)

where the second inequality follows from (iii). In view of (i), (4.8) implies that φ( f (S,T , u)− f (T ,T ,u)) 0, which because of the properties of φ, reduces to f (S,T ,u)− f (T ,T , u) 0. But f (T ,T ,u) = 0 and hence f (S,T ,u) 0, which is precisely (3.15). Therefore, the rest of the proof is identical to that of part (a) of Theorem 3.1.

(b)–(f) The proofs are similar to that of part (a).

Theorem 4.2 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following six sets of hypotheses is satisfied:

(a) (i) for each i∈ I+ ≡ I+(u), fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, and φi(0)= 0;

1120 Multiobjective fractional subset programming

(ii) for each j ∈ J+ ≡ J+(v), Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing,

and φ j(0)= 0; (iii) ρ +

∑ j∈J+ vj ρ j 0, where ρ =∑i∈I+ uiρi;

(b) (i) for each i∈ I+, fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increas- ing, and φi(0)= 0;

(ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0; (iii) ρ + ρ 0;

(c) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) for each j ∈ J+, Hj is strictly (,b, φ j , ρ j ,θ)-pseudounivex at T , φ j is increasing,

and φ j(0)= 0; (iii) ρ +

∑ j∈J+ vj ρ j 0;

(d) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) h(·,v) is strictly (,b, φ, ρ,θ)-pseudounivex at T , φ is increasing, and φ(0)= 0; (iii) ρ + ρ 0;

(e) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) for each j ∈ J+, Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing, and

φ j(0)= 0; (iii) ρ +

∑ j∈J+ vj ρ j > 0;

(f) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0; (iii) ρ + ρ > 0;

Then (S) ψ(T ,u,v).

Proof. (a) Suppose to the contrary that (S) ψ(T ,u,v). This implies that for each i ∈ p, Fi(S)/Gi(S) Fi(T)/Gi(T), and so Gi(T)Fi(S)− Fi(T)Gi(S) 0, with strict inequality holding for at least one index ∈ p. Hence for each i∈ p,

Gi(T)Fi(S)−Fi(T)Gi(S) 0=Gi(T)Fi(T)−Fi(T)Gi(T), (4.9)

which in view of the properties of φi can be expressed as φi( fi(S,T)− fi(T ,T)) 0. By virtue of (i), these inequalities imply that for each i∈ I+,

( S,T ;b

]) <−ρid2(θ(S,T)

) . (4.10)

) . (4.12)

But this contradicts (4.7), which is valid for the present case because of our hypotheses set forth in (ii). Hence (S) ψ(T ,u,v).

(b)–(f) The proofs are similar to that of part (a).

The next theorem may be viewed as a variant of Theorem 4.2; its proof is similar to that of Theorem 4.2 and hence omitted.

Theorem 4.3 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following four sets of hypotheses is satis- fied:

(a) (i) for each i∈ I1+ = ∅, fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i∈ I2+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , and for each i∈ I+ ≡ I+(u), φi is increasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

(ii) for each j ∈ J+ ≡ J+(v), Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing

and φ j(0)= 0; (iii) ρ +

∑ j∈J+ vj ρ j 0, where ρ =∑i∈I+ uiρi;

(b) (i) for each i∈ I1+ = ∅, fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i ∈ I2+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , and for each i ∈ I+, φi is in- creasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

(ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0; (iii) ρ + ρ 0;

(c) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) for each j ∈ J1+ = ∅,Hj is strictly (,b, φ j , ρ j ,θ)-pseudounivex at T , for each j ∈ J2+, Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , and for each j ∈ J+, φ j is increasing

and φ j(0)= 0, where {J1+, J2+} is a partition of J+; (iii) ρ +

∑ j∈J+ vj ρ j 0;

(d) (i) for each i∈ I1+, fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i∈ I2+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , and for each i∈ I+, φi is increasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

(ii) for each j ∈ J1+, Hj is strictly (,b, φ j , ρ j ,θ)-pseudounivex at T , for each j ∈ J2+,

Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , and for each j ∈ J+, φ j is increasing and

φ j(0)= 0, where {J1+, J2+} is a partition of J+; (iii) ρ +

∑ j∈J+ vj ρ j 0;

(iv) I1+ = ∅, J1+ = ∅, or ρ + ∑

j∈J+ vj ρ j > 0. Then (x) ψ(T ,u,v).

Theorem 4.4 (strong duality). Let S∗ be a regular efficient solution of (P), let (S,S∗; DF(S∗))=∑n

k=1DkF(S∗),χSk − χS∗k for any differentiable function F :An→R and S∈An,

1122 Multiobjective fractional subset programming

and assume that any one of the sixteen sets of hypotheses specified in Theorems 4.1, 4.2, and 4.3 holds for all feasible solutions of (DII). Then there exist u∗ ∈U and v∗ ∈Rq

+ such that (S∗,u∗,v∗) is an efficient solution of (DII) and φ(S∗)= ψ(S∗,u∗,v∗).

Proof. By Theorem 2.9, there exist u∗ ∈ U and v∗ ∈ Rq + such that (S∗,u∗,v∗) is a fea-

sible solution of (DII) and (S∗)= ψ(S∗,u∗,v∗). That (S∗,u∗,v∗) is efficient for (DII) follows from (the corresponding part of) Theorems 4.1, 4.2, and 4.3.

The proofs of the next two theorems are similar to that of Theorem 3.3.

Theorem 4.5 (strict converse duality). Let S∗ and (S,S∗;·) be as in Theorem 4.4, let (S, u, v) be a feasible solution of (DII) such that f (S∗, S, u) 0, and assume that either one of the two sets of hypotheses specified in parts (a) and (b) of Theorem 4.1 is satisfied for the fea- sible solution (S, u, v) of (DII). Assume, furthermore, that f (·, S, u) is strictly (,b, φ, ρ,θ)- pseudounivex at S and that φ(a) > 0⇒ a > 0. Then S= S∗, that is, S is an efficient solution of (P).

Theorem 4.6 (strict converse duality). Let S∗ and (S,S∗;·) be as in Theorem 4.4, let (S, u, v) be a feasible solution of (DII) such that f (S∗, S, u) 0, and assume that any one of the four sets of hypotheses specified in parts (c)–(f) of Theorem 4.1 is satisfied for the feasible solution (S, u, v) of (DII). Assume, furthermore, that f (·, S, u) is (,b, φ, ρ,θ)-quasiunivex at S and that φ(a) > 0⇒ a > 0. Then S= S∗, that is, S is an efficient solution of (P).

5. Duality model III

In this section, we formulate a more general duality model for (P) with a more flexible structure which will allow us to establish duality under various generalized (,b,φ,ρ,θ)- univexity hypotheses that can be imposed on certain combinations of the problem func- tions. This will be accomplished by utilizing a partitioning scheme which was originally proposed in [2] for the purpose of constructing generalized dual problems for nonlin- ear programs with point-functions. Prior to formulating this duality model, we need to introduce some additional notation.

Let {J0, J1, . . . , Jm} be a partition of the index set q; thus Jr ⊂ q for each r ∈ {0,1, . . . ,m}, Jr ∩ Js = ∅ for each r,s ∈ {0,1, . . . ,m} with r = s, and ∪m

r=0Jr = q. In addition, we will make use of the real-valued functions Πi(·,T ,v), Π(·,T ,u,v), and Λt(·,v) defined, for fixed T , u, and v, on An by

Πi(R,T ,v)=Gi(T)

ui

{ Gi(T)

[ Fi(R) +

(5.1)

Maximize (F1(T) +

T ∈An, u∈U0, v ∈Rq +, (5.5)

where (S,T ;·) : Ln1(X ,A,µ)→R is a sublinear function. We next show that (DIII) is a dual problem for (P) by proving weak and strong duality

theorems.

Theorem 5.1 (weak duality). Let S and (T ,u,v), with u > 0, be arbitrary feasible solu- tions of (P) and (DIII), respectively, and assume that any one of the following four sets of hypotheses is satisfied:

(a) (i) Π(·,T ,u,v) is (,b, φ, ρ,θ) -pseudounivex at T , and φ(a) 0⇒ a 0; (ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and

φt(0)= 0; (iii) ρ+

∑m t=1 ρt 0;

(b) (i) Π(·,T ,u,v) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each t ∈m, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , φt is increas-

ing, and φt(0)= 0; (iii) ρ+

∑m t=1 ρt 0;

(c) (i) Π(·,T ,u,v) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and

φt(0)= 0; (iii) ρ+

∑m t=1 ρt > 0;

(d) (i) Π(·,T ,u,v) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each t ∈m1, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and

φt(0)= 0, for each t ∈m2 = ∅, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at

T , φt is increasing, and φt(0)= 0, where {m1, m2} is a partition of m; (iii) ρ+

∑m t=1 ρt 0.

Then (S) ω(T ,u,v), where ω = (ω1, . . . ,ωp) is the objective function of (DIII).

1124 Multiobjective fractional subset programming

) 0.

(5.6)

Since S∈ F and v 0, it is clear from (5.4) that for each t ∈m,

Λt(S,v)= ∑ j∈Jt

v jHj(T)=Λt(T ,v), (5.7)

and so using the properties of φt, we get φt ( Λt(S,v)−Λt(T ,v)

) 0, which in view of (ii)

implies that for each t ∈m,

) −ρtd2(θ(S,T)

) . (5.8)

(5.10)

where the second inequality follows from (iii). Because of (i), this inequality implies that φ(Π(S,T ,u,v)−Π(T ,T ,u,v)) 0. But φ(a) ≥ 0 ⇒ a ≥ 0, and so we get Π(S,T ,u,v) Π(T ,T ,u,v) = 0. Inasmuch as S ∈ F, u > 0, v 0, and Gi(T) > 0, i ∈ p, this inequality yields

p∑ i=1

] Gi(S)

} 0. (5.11)

Since u > 0, this inequality implies that( G1(T)F1(S)− [F1(T) +Λ0(T ,v)

] G1(S), . . . ,Gp(T)Fp(S)

(S)= ( F1(S) G1(S)

(b)–(d) The proofs are similar to that of part (a).

Theorem 5.2 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DIII), respectively, and assume that any one of the following six sets of hypotheses is satis- fied:

(a) (i) for each i∈ I+ ≡ I+(u), Πi(·,T ,v, ) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, and φi(0)= 0;

(ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt 0;

(b) (i) for each i ∈ I+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , φi is in- creasing, and φi(0)= 0;

(ii) for each t ∈m, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , φt is increas- ing, and φt(0)= 0;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt 0;

(c) (i) for each i ∈ I+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , φi is in- creasing, and φi(0)= 0;

(ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt > 0;

(d) (i) for each i ∈ I1+ = ∅, Πi(·,T ,v) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i∈ I2+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , and for each i∈ I+, φi is increasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

(ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt 0;

(e) (i) for each i ∈ I+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , φi is in- creasing, and φi(0)= 0;

(ii) for each t ∈m1 = ∅, Λt(·,v) is strictly (,b, φt , ρt,θ)-pseudounivex at T , φt is

increasing, and φt(0) = 0, and for each t ∈m2, Λt(·,v) is (,b, φt, ρt,θ)-quasi-

univex at T , and for each t ∈m, φt is increasing and φt(0)= 0, where {m1, m2} is a partition of m;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt 0;

(f) (i) for each i ∈ I1+, Πi(·,T ,v) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i∈ I2+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , and for each i∈ I+, φi is increasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

1126 Multiobjective fractional subset programming

(ii) for each t ∈ m1, Λt(·,v) is strictly (,b, φt , ρt,θ)-pseudounivex at T , for each

t ∈m2, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , and for each t ∈m, φt is in-

creasing and φt(0)= 0, where {m1,m2} is a partition of m; (iii)

∑ i∈I+ uiρi +

(iv) I1+ = ∅, m1 = ∅, ∑

t=1 ρt > 0. Then (S) ω(T ,u,v).

Proof. (a) Suppose to the contrary that (S) ω(T ,u,v). This implies that for each i∈ p, Gi(T)Fi(S)− [Fi(T) +Λ0(T ,v)]Gi(S) 0, with strict inequality holding for at least one index ∈ p. Using these inequalities along with the primal feasibility of S and nonnega- tivity of v, we see that

Πi(S,T ,v)=Gi(T)

=Πi(T ,T ,v).

(5.14)

(5.15)

) , (5.17)

where the second inequality follows from (iii). Obviously, this inequality contradicts (5.9), which is valid for the present case because of (ii). Hence we must have (S) ω(T ,u,v).

(b)–(f) The proofs are similar to that of part (a).

G. J. Zalmai 1127

Theorem 5.3 (strong duality). Let S∗ be a regular efficient solution of (P), let (S,S∗; DF(S∗))=∑n

k=1DkF(S∗),χSk − χS∗k for any differentiable function F :An→R and S∈An, and assume that any one of the ten sets of hypotheses specified in Theorems 5.1 and 5.2 holds for all feasible solutions of (DIII). Then there exist u∗ ∈ U and v∗ ∈ Rq

+ such that (S∗,u∗,v∗) is an efficient solution of (DIII) and (S∗)= ω(S∗,u∗,v∗).

Proof. By Theorem 2.9, there exist u∗ ∈U and v ∈Rq + such that

⟨ p∑ i=1

] +

(5.18)

v jHj(S∗)= 0, j ∈ q. (5.19)

Now if we let v∗j = v j /Gi(S∗) for each j ∈ J0, v∗j = v j for each j ∈ q \ J0, and observe that Λ0(S∗,v∗)= 0, then (5.18) and (5.19) can be rewritten as follows:

⟨ p∑ i=1

0 ∀Sk ∈A, k ∈ n,

(5.20)∑ j∈Jt

v∗j Hj(S∗)= 0, t ∈m. (5.21)

From (5.20) and (5.21) it is clear that (S∗,u∗,v∗) is a feasible solution of (DIII). Proceed- ing as in the proof of Theorem 3.3, it can easily be verified that it is an efficient solution of (DIII).

We next show that certain modifications in Theorem 5.1 lead to a number of strict converse duality results for (P) and (DIII).

Theorem 5.4 (strict converse duality). Let S∗ and be as in Theorem 5.3, let (S, u, v) be a feasible solution of (DIII) such that

p∑ i=1

] Gi(S∗)

} 0, (5.22)

and assume that any one of the four sets of hypotheses specified in Theorem 5.1 is satisfied for the feasible solution (S, u, v) of (DIII). Assume furthermore that any one of the following corresponding conditions holds:

(a) Π(·, S, u, v) is strictly (,b, φ, ρ,θ)-pseudounivex at S, and φ(a) > 0⇒ a > 0; (b) Π(·, S, u, v) is (,b, φ, ρ,θ)-quasiunivex at S, and φ(a) > 0⇒ a > 0;

1128 Multiobjective fractional subset programming

(c) Π(·, S, u, v) is (,b, φ, ρ,θ)-quasiunivex at S, and φ(a) > 0⇒ a > 0; (d) Π(·, S, u, v) is (,b, φ, ρ,θ)-quasiunivex at S, and φ(a) > 0⇒ a > 0.

Then S= S∗.

Proof. The proof is similar to that of Theorem 3.3.

Evidently, (DIII) contains a number of important special cases which can easily be identified by appropriate choices of the partitioning sets J0, J1, . . . , Jm, and the sublinear function (S,T ;·). We conclude this section by briefly looking at a few of these special cases. In each case, we specify the required conditions for duality by specializing part (a) of Theorem 5.2.

If we let J0 = q, then (DIII) takes the following form: (DIIIa)

Maximize (F1(T) +

(5.24)

(DIIIa) is dual to (P) if for each i∈ I+, the function

R−→Gi(T)

] Gi(R) (5.25)

is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0)= 0, and ∑

i∈I+ uiρi 0. If we let J1 = q, then (DIII) becomes

(DIIIb)

G. J. Zalmai 1129

(DIIIb) is dual to (P) if for each i ∈ I+, the function R→ Gi(T)Fi(R)− Fi(T)Gi(R) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0) = 0, the function R→∑q

j=1 vjHj(R) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, φ(0)= 0, and ∑

i∈I+ uiρi + ρ 0.

If we choose J0 =∅, then we obtain the following special case of (DIII): (DIIIc)

Maximize ( F1(T) G1(T)

v jHj(T) 0, t ∈m. (5.29)

(DIIIc) is dual to (P) if for each i ∈ I+, the function R→ Gi(T)Fi(R)− Fi(T)Gi(R) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0) = 0, for each t ∈ m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, φt(0) = 0, and

∑ i∈I+ uiρi +∑m

t=1 ρt 0. If we set m= q, Jt = {t}, t ∈ q, then (DIII) reduces to the following problem:

(DIIId)

vjHj(T) 0, j ∈ q. (5.31)

(DIIId) is dual to (P) if for each i∈ I+, the function R→ Gi(T)Fi(R)− Fi(T)Gi(R) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0)= 0, for each j ∈ q, R→ vjHj(R) is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing, φ j(0) = 0, and

∑ i∈I+ uiρi +∑q

j=1 ρ j 0. If we let Jt =∅, t = 2,3, . . . ,m, then we get the following special case of (DIII):

(DIIIe)

subject to (5.5) and

vjHj(T) 0. (5.34)

(DIIIe) is dual to (P) if for each i∈ I+, the function

R−→Gi(T)

] Gi(R) (5.35)

is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0)= 0, the function R→∑ j∈J1 vjHj(R) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, φ(0)= 0, and

∑ i∈I+ uiρi +

ρ 0. In a similar manner, one can obtain a vast number of duality theorems for (P) by

specializing the other nine sets of conditions for (DIIIa)–(DIIIe) and other special cases of (DIII).

6. Duality model IV

In this section, we present another general duality model for (P) that is different from (DIII) in that here in constructing the constraints we not only use a partition of the index set q, but also a partition of the set p. A parametric point-function version of this dual problem was considered earlier in [4].

Let {I0,I1, . . . ,Ik} be a partition of p such that K ≡ {0,1, . . . ,k} ⊂M ≡ {0,1, . . . ,m}, and let the function t(·,S,u,v) :An→R be defined, for fixed S, u, and v, by

t(T ,S,u,v)= ∑ i∈It

Maximize ( F1(T) G1(T)

T ∈An, u∈U , v ∈Rq +, (6.5)

where (S,T ;·) : Ln1(X ,A,µ)→R is a sublinear function. The next two theorems show that (DIV) is a dual problem for (P).

Theorem 6.1 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DIV), respectively, and assume that any one of the following six sets of hypotheses is satis- fied:

(a) (i) for each t ∈ K , t(·,T ,u,v) is strictly (,b,φt,ρt,θ)-pseudounivex at T , φt is increasing, and φt(0)= 0;

(ii) for each t ∈M \K , Λt(·,v) is (,b,φt,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

t∈M ρt 0;

(b) (i) for each t ∈ K , t(·,T ,u,v) is prestrictly (,b,φt ,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(ii) for each t ∈M \K , Λt(·,v) is strictly (,b,φt,ρt,θ)-pseudounivex at T , φt is in- creasing, and φt(0)= 0;

(iii) ∑

t∈M ρt 0;

(c) (i) for each t ∈ K , t(·,T ,u,v) is prestrictly (,b,φt ,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(ii) for each t ∈M \K , Λt(·,v) is (,b,φt,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

t∈M ρt > 0;

(d) (i) for each t ∈ K1 = ∅, t(·,T ,u,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , for each t ∈ K2, t(·,T ,u,v) is prestrictly (,b, φt , ρt,θ)-quasiunivex at T , and for each t ∈ K , φt is increasing and φt(0)= 0, where {K1,K2} is a partition of K ;

(ii) for each t ∈M \K , Λt(·,v) is (,b,φt,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

t∈M ρt 0;

(e) (i) for each t ∈ K , t(·,T ,u,v) is prestrictly (,b,φt ,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(ii) for each t ∈ (M \K)1 = ∅, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , for each t ∈ (M \K)2,Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex atT , and for each t ∈ M \K , φt is increasing and φt(0)= 0, where {(M \K)1, (M \K)2} is a partition of M \K ;

(iii) ∑

1132 Multiobjective fractional subset programming

(f) (i) for each t ∈ K1, t(·,T ,u,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , for each t ∈ K2, t(·,T ,u,v) is prestrictly (,b, φt, ρt,θ)-quasiunivex at T , and for each t ∈ K , φt is increasing and φt(0)= 0, where {K1,K2} is a partition of K ;

(ii) for each t ∈ (M \K)1, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , φt is increasing, and φt(0) = 0, and for each t ∈ (M \K)2, Λt(·,v) is (,b, φt, ρt,θ)- quasiunivex at T , and for each t ∈M \K , φt is increasing and φt(0) = 0, where {(M \K)1,(M \K)2} is a partition of M \K ;

(iii) ∑

∑ t∈M ρt > 0.

Then (S) δ(T ,u,v), where δ = (δ1, . . . ,δp) is the objective function of (DIV).

(6.6)

(6.7)

) −ρtd2(θ(S,T)

) . (6.8)

G. J. Zalmai 1133

) 0,

(6.10)

which contradicts (6.3). Therefore, (S) δ(T ,u,v). (b)–(f) The proofs are similar to that of part (a).

Theorem 6.2 (strong duality). Let S∗ be a regular efficient solution of (P), let (S,S∗; DF(S∗)) =∑n

k=1DkF(S∗),χSk − χS∗k for any differentiable function F : An → R and S ∈ An, and assume that any one of the six sets of hypotheses specified in Theorem 6.1 holds for all feasible solutions of (DIV). Then there exist u∗ ∈U and v∗ ∈Rq

+ such that (S∗,u∗,v∗) is an efficient solution of (DIV) and (S∗)= δ(S∗,u∗,v∗).

Proof. The proof is similar to that of Theorem 4.4.

Various special cases of (DIV) can easily be generated by appropriate choices of the partitioning sets I0,I1, . . . ,Ik, J0, and (S,T ;·).

References

[1] H. W. Corley, Optimization theory for n-set functions, J. Math. Anal. Appl. 127 (1987), no. 1, 193–205.

[2] B. Mond and T. Weir, Generalized concavity and duality, Generalized Concavity in Optimization and Economics (S. Schaible and W. T. Ziemba, eds.), Academic Press, New York, 1981, pp. 263–279.

[3] R. J. T. Morris, Optimal constrained selection of a measurable subset, J. Math. Anal. Appl. 70 (1979), no. 2, 546–562.

[4] X. M. Yang, Generalized convex duality for multiobjective fractional programs, Opsearch 31 (1994), no. 2, 155–163.

[5] G. J. Zalmai, Efficiency conditions and duality models for multiobjective fractional subset pro- gramming problems with generalized (,α,ρ,θ)-V-convex functions, Comput. Math. Appl. 43 (2002), no. 12, 1489–1520.

[6] , Generalized (,b,φ,ρ,θ)-univex n-set functions and global semiparametric sufficient efficiency conditions in multiobjective fractional subset programming, to appear in Int. J. Math. Math. Sci.

G. J. Zalmai: Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI 49855, USA

E-mail address: [email protected]

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Received 28 December 2003 and in revised form 3 October 2004

We construct a number of semiparametric duality models and establish appropriate du- ality results under various generalized (,b,φ,ρ,θ)-univexity assumptions for a multiob- jective fractional subset programming problem.

1. Introduction

In this paper, we will present a number of semiparametric duality results under various generalized (,b,φ,ρ,θ)-univexity hypotheses for the following multiobjective fractional subset programming problem:

(P)

) subject to Hj(S) 0, j ∈ q, S∈An, (1.1)

whereAn is the n-fold product of the σ-algebraA of subsets of a given set X ,Fi,Gi, i∈ p ≡ {1,2, . . . , p}, and Hj , j ∈ q, are real-valued functions defined on An, and for each i ∈ p, Gi(S) > 0 for all S∈An such that Hj(S) 0, j ∈ q.

This paper is essentially a continuation of the investigation that was initiated in the companion paper [6] where some information about multiobjective fractional program- ming problems involving point-functions as well as n-set functions was presented, a fairly comprehensive list of references for multiobjective fractional subset programming prob- lems was provided, a brief overview of the available results pertaining to multiobjective fractional subset programming problems was given, and numerous sets of semiparamet- ric sufficient efficiency conditions under various generalized (,b,φ,ρ,θ)-univexity as- sumptions were established. These and some other related material that were discussed in [6] will not be repeated in the present paper. Making use of the semiparametric sufficient efficiency criteria developed in [6] in conjunction with a certain necessary efficiency re- sult that will be recalled in the next section, here we will construct several semiparametric duality models for (P) with varying degrees of generality and, in each case, prove appro- priate weak, strong, and strict converse duality theorems under a number of generalized (,b,φ,ρ,θ)-univexity conditions.

Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:7 (2005) 1109–1133 DOI: 10.1155/IJMMS.2005.1109

1110 Multiobjective fractional subset programming

The rest of this paper is organized as follows. In Section 3 we consider a simple dual problem and prove weak, strong, and strict converse duality theorems. In Section 4 we formulate another dual problem with a relatively more flexible structure that allows for a greater variety of generalized (,b,φ,ρ,θ)-univexity conditions under which duality can be established. In Sections 5 and 6 we state and discuss two general duality models which are, in fact, two families of dual problems for (P), whose members can easily be identified by appropriate choices of certain sets and functions.

Evidently, all of these duality results are also applicable, when appropriately special- ized, to the following three classes of problems with multiple, fractional, and conventional objective functions, which are particular cases of (P):

(P1)

F1(S), (1.4)

where F (assumed to be nonempty) is the feasible set of (P), that is,

F= {S∈An :Hj(S) 0, j ∈ q}. (1.5)

Since in most cases the duality results established for (P) can easily be modified and restated for each one of the above problems, we will not explicitly state these results.

2. Preliminaries

In this section, we gather, for convenience of reference, a few basic definitions and auxil- iary results which will be used frequently throughout the sequel.

Let (X ,A,µ) be a finite atomless measure space with L1(X ,A,µ) separable, and let d be the pseudometric on An defined by

d(R,S)= [ n∑ i=1

)∈An, (2.1)

where denotes symmetric difference; thus (An,d) is a pseudometric space. For h ∈ L1(X ,A,µ) and T ∈ A with characteristic function χT ∈ L∞(X ,A,µ), the integral

∫ T hdµ

will be denoted by h,χT. We next define the notion of differentiability for n-set functions. It was originally in-

troduced by Morris [3] for a set function, and subsequently extended by Corley [1] for n-set functions.

G. J. Zalmai 1111

Definition 2.1. A function F : A → R is said to be differentiable at S∗ if there exists DF(S∗)∈ L1(X ,A,µ), called the derivative of F at S∗, such that for each S∈A,

F(S)= F(S∗)+ ⟨ DF

( S∗ ) ,χS− χS∗

where VF(S,S∗) is o(d(S,S∗)), that is, limd(S,S∗)→0VF(S,S∗)/d(S,S∗)= 0.

Definition 2.2. A function G : An → R is said to have a partial derivative at S∗ = (S∗1 , . . . , S∗n )∈An with respect to its ith argument if the function F(Si)=G(S∗1 , . . . ,S∗i−1,Si,S∗i+1, . . . , S∗n ) has derivativeDF(S∗i ), i∈ n; in that case, the ith partial derivative ofG at S∗ is defined to be DiG(S∗)=DF(S∗i ), i∈ n.

Definition 2.3. A function G : An → R is said to be differentiable at S∗ if all the partial derivatives DiG(S∗), i∈ n, exist and

G(S)=G(S∗)+ n∑ i=1

⟨ DiG

⟩ +WG

( S,S∗

) , (2.3)

where WG(S,S∗) is o(d(S,S∗)) for all S∈An. We next recall the definitions of the generalized (,b,φ,ρ,θ)-univex n-set functions

which will be used in the statements of our duality theorems. For more information about these and a number of other related classes of n-set functions, the reader is referred to [6]. We begin by defining a sublinear function which is an integral part of all the subsequent definitions.

Definition 2.4. A function : Rn → R is said to be sublinear (superlinear) if (x + y) ()(x) + (y) for all x, y ∈Rn, and (ax)= a(x) for all x ∈Rn and a∈R+ ≡ [0,∞).

Let S,S∗ ∈An, and assume that the function F :An→R is differentiable at S∗.

Definition 2.5. The function F is said to be (strictly) (,b,φ,ρ,θ)-univex at S∗ if there exist a sublinear function (S,S∗;·) : Ln1(X ,A,µ)→ R, a function b : An ×An → R with positive values, a function θ : An ×An → An ×An such that S = S∗ ⇒ θ(S,S∗) = (0,0), a function φ :R→R, and a real number ρ such that for each S∈An,

φ ( F(S)−F(S∗))(>)

+ ρd2(θ(S,S∗ )) . (2.4)

Definition 2.6. The function F is said to be (strictly) (,b,φ,ρ,θ)-pseudounivex at S∗ if there exist a sublinear function (S,S∗;·) : Ln1(X ,A,µ) → R, a function b : An ×An → R with positive values, a function θ : An ×An → An ×An such that S = S∗ ⇒ θ(S,S∗) = (0,0), a function φ :R→R, and a real number ρ such that for each S∈An (S = S∗),

( S,S∗;b

Definition 2.7. The function F is said to be (prestrictly) (,b,φ,ρ,θ)-quasiunivex at S∗

if there exist a sublinear function (S,S∗;·) : Ln1(X ,A,µ)→ R, a function b : An ×An → R with positive values, a function θ : An ×An → An ×An such that S = S∗ ⇒ θ(S,S∗) = (0,0), a function φ :R→R, and a real number ρ such that for each S∈An,

φ ( F(S)−F(S∗))(<) 0=⇒

1112 Multiobjective fractional subset programming

From the above definitions it is clear that if F is (,b,φ,ρ,θ)-univex at S∗, then it is both (,b,φ,ρ,θ)-pseudounivex and (,b,φ,ρ,θ)-quasiunivex at S∗, if F is (,b,φ,ρ,θ)- quasiunivex at S∗, then it is prestrictly (,b,φ,ρ,θ)-quasiunivex at S∗, and if F is strictly (,b,φ,ρ,θ)-pseudounivex at S∗, then it is (,b,φ,ρ,θ)-quasiunivex at S∗.

In the proofs of the duality theorems, sometimes it may be more convenient to use certain alternative but equivalent forms of the above definitions. These are obtained by considering the contrapositive statements. For example, (,b,φ,ρ,θ)-quasiunivexity can be defined in the following equivalent way: F is said to be (,b,φ,ρ,θ)-quasiunivex at S∗

if for each S∈An,

( S,S∗;b

))=⇒ φ ( F(S)−F(S∗)) > 0. (2.7)

Needless to say, the new classes of generalized convex n-set functions specified in Def- initions 2.5, 2.6, and 2.7 contain a variety of special cases; in particular, they subsume all the previously defined types of generalized n-set functions. This can easily be seen by appropriate choices of , b, φ, ρ, and θ.

In the sequel we will also need a consistent notation for vector inequalities. For all a,b ∈ Rm, the following order notation will be used: a b if and only if ai bi for all i∈m; a b if and only if ai bi for all i∈m, but a = b; a > b if and only if ai > bi for all i∈m; a b is the negation of a b.

Throughout the sequel we will deal exclusively with the efficient solutions of (P). An x∗ ∈ is said to be an efficient solution of (P) if there is no other x ∈ such that (x) (x∗), where is the objective function of (P).

Next, we recall a set of parametric necessary efficiency conditions for (P).

Theorem 2.8 [5]. Assume that Fi,Gi, i ∈ p, and Hj , j ∈ q, are differentiable at S∗ ∈ An,

and that for each i∈ p, there exist Si ∈An such that

Hj ( S∗ )

⟨ DkF

⟩ < 0. (2.9)

If S∗ is an efficient solution of (P) and λ∗i = (S∗), i ∈ p, then there exist u∗ ∈ U = {u ∈ Rp : u > 0,

∑p i=1ui = 1} and v∗ ∈Rq

+ such that

⟩ 0, ∀S∈An,

G. J. Zalmai 1113

The above theorem contains two sets of parameters u∗i and λ∗i , i ∈ p, which were in- troduced as a consequence of our indirect approach in [5] requiring two intermediate auxiliary problems. It is possible to eliminate one of these two sets of parameters and thus obtain a semiparametric version of Theorem 2.8. Indeed, this can be accomplished by simply replacing λ∗i by Fi(S∗)/Gi(S∗), i ∈ p, and redefining u∗ and v∗. For future reference, we state this in the next theorem.

Theorem 2.9. Assume that Fi,Gi, i ∈ p, and Hj , j ∈ q, are differentiable at S∗ ∈ An, and

that for each i∈ p, there exist Si ∈An such that

Hj ( S∗ )

n∑ k=1

⟨ Gi ( S∗ ) DkF

( S∗ )−Fi(S∗)DkG

⟩ < 0. (2.12)

If S∗ is an efficient solution of (P), then there exist u∗ ∈U and v∗ ∈Rq + such that

n∑ k=1

⟨ p∑ i=1

( S∗ )−Fi(S∗)DkGi

⟩ 0, ∀S∈An,

v∗j Hj ( S∗ )= 0, j ∈ q.

For simplicity, we will henceforth refer to an efficient solution S∗ of (P) satisfying (2.11) and (2.12) for some Si, i∈ p, as a normal efficient solution.

The form and contents of the necessary efficiency conditions given in Theorem 2.9 in conjunction with the sufficient efficiency results established in [6] provide clear guide- lines for constructing various types of semiparametric duality models for (P).

3. Duality model I

In this section, we discuss a duality model for (P) with a somewhat restricted constraint structure that allows only certain types of generalized (,b,φ,ρ,θ)-univexity conditions for establishing duality. More general duality models will be presented in subsequent sec- tions.

In the remainder of this paper, we assume that the functions Fi,Gi, i∈ p, andHj , j ∈ q, are differentiable onAn and that Fi(T) 0 andGi(T) > 0 for each i∈ p and for all T such that (T ,u,v) is a feasible solution of the dual problem under consideration.

1114 Multiobjective fractional subset programming

Consider the following problem: (DI)

Minimize ( F1(T) G1(T)

T ∈An, u∈U , v ∈Rq +, (3.4)

where (S,T ;·) : Ln1(X ,A,µ)→R is a sublinear function. The following two theorems show that (DI) is a dual problem for (P).

Theorem 3.1 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DI), respectively, and assume that any one of the following three sets of hypotheses is satis- fied:

(a) (i) for each i ∈ p, Fi is (,b, φ, ρi,θ)-univex at T , and −Gi is (,b, φ, ρi,θ)-univex

at T , φ is superlinear, and φ(a) 0⇒ a 0; (ii) for each j ∈ q, Hj is (,b, φ, ρ,θ)-univex at T , φ is increasing, and φ(0)= 0;

(iii) ρ∗ + ∑

j∈J+ vj ρ j 0, where ρ∗ =∑p i=1ui[Gi(T)ρi +Fi(T)ρi];

(b) (i) for each i ∈ p, Fi is (,b, φ, ρi,θ)-univex at T , and −Gi is (, b, φ, ρi,θ)-univex

at T , φ is superlinear, and φ(a) 0⇒ a 0; (ii) the function T →∑q

j=1 vjHj(T) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing,

and φ(0)= 0; (iii) ρ∗ + ρ 0;

(c) (i) the Lagrangian-type function

T −→ p∑ i=1

vjHj(T), (3.5)

where S is fixed in An, is (,b, φ,0,θ)-pseudounivex at T and φ(a) 0⇒ a 0. Then (S) ξ(T ,u,v), where ξ = (ξ1, . . . ,ξp) is the objective function of (DI).

Proof. (a) From (i) and (ii) it follows that

φ ( Fi(S)−Fi(T)

G. J. Zalmai 1115

Multiplying (3.6) by uiGi(T) and (3.7) by uiFi(T), i∈ p, adding the resulting inequalities,

and then using the superlinearity of φ and sublinearity of (S,T ;·), we obtain

φ

φ

(3.10)

Since v 0, S∈ F, and (3.3) holds, it is clear that

q∑ j=1

vj [ Hj(S)−Hj(T)

] 0, (3.11)

which implies, in view of the properties of φ, that the left-hand side of (3.10) is less than or equal to zero, that is,

0

vjDHj(T)

) 0.

(3.13)

Now adding (3.9) and (3.12), and then using (3.13) and (iii), we obtain

φ

p∑ i=1

ui [ Gi(T)Fi(S)−Fi(T)Gi(S)

Since u > 0, (3.15) implies that( G1(T)F1(S)−F1(T)G1(S), . . . ,Gp(T)Fp(S)−Fp(T)Gp(S)

) (0, . . . ,0), (3.16)

(S)= ( F1(S) G1(S)

) = ξ(T ,u,v). (3.17)

(b) Since for each j ∈ q, vjHj(S) 0, it follows from (3.3) that

q∑ j=1

vjHj(T), (3.18)

φ

vjDHj(T)

) −ρd2(θ(S,T)

) . (3.20)

Now combining (3.9), (3.13), and (3.20), and using (iii), we obtain (3.15). Therefore, the rest of the proof is identical to that of part (a).

(c) From the (,b, φ,0,θ)-pseudounivexity assumption and (3.2) it follows that

φ

In view of the properties of φ, this inequality becomes

p∑ i=1

ui [ Gi(T)Fi(S)−Fi(T)Gi(S)

vjHj(T) 0, (3.22)

which because of (3.3), primal feasibility of S, and nonnegativity of v, reduces to (3.15), and so the rest of the proof is identical to that of part (a).

Theorem 3.2 (strong duality). Let S∗ be a regular efficient solution of (P), let (S,S∗; DF(S∗)) =∑n

k=1DkF(S∗),χSk − χS∗k for any differentiable function F : An → R and S ∈ An, and assume that any one of the three sets of hypotheses specified in Theorem 3.1 holds for all feasible solutions of (DI). Then there exist u∗ ∈U and v∗ ∈Rq

+ such that (S∗,u∗,v∗) is an efficient solution of (DI) and (S∗)= ξ(S∗,u∗,v∗).

G. J. Zalmai 1117

Proof. By Theorem 2.9, there exist u∗ ∈U and v∗ ∈Rq + such that (S∗,u∗,v∗) is a feasible

solution of (DI). If it were not an efficient solution, then there would exist a feasible solution (T , u, v) such that ξ(T , u, v) ξ(S∗,u∗,v∗)= (S∗), which contradicts the weak duality relation established in Theorem 5.1. Therefore, (S∗,u∗,v∗) is an efficient solution of (DI).

We also have the following converse duality result for (P) and (DI).

Theorem 3.3 (strict converse duality). Let S∗ and (S,S∗;·) be as in Theorem 3.2, let (S, u, v) be a feasible solution of (DI) such that

p∑ i=1

ui [ Gi(S)Fi(S∗)−Fi(S)Gi(S∗)

] 0. (3.23)

Furthermore, assume that any one of the following three sets of hypotheses is satisfied: (a) the assumptions specified in part (a) of Theorem 3.1 are satisfied for the feasible so-

lution (S, u, v) of (DI); Fi is strictly (,b, φ, ρi,θ)-univex at S for at least one index i ∈ p with the corresponding component ui of u positive, and φ(a) > 0⇒ a > 0, or

−Gi is strictly (,b, φ, ρi,θ)-univex at S for at least one index i ∈ p with ui posi-

tive, and φ(a) > 0⇒ a > 0, or Hj is strictly (,b, φ, ρ j ,θ)-univex at S for at least one

index j ∈ q with v j positive, and φ(a) > 0⇒ a > 0, or ∑p

i=1 ui[Gi(S)ρi + Fi(S)ρi] +∑q j=1 v j ρ j > 0;

(b) the assumptions specified in part (b) of Theorem 3.1 are satisfied for the feasible solu- tion (S, u, v) of (DI), Fi and φ or−Gi and φ satisfy the requirements described in part (a), or the function R→∑q

j=1 v jHj(R) is strictly (,b, φ, ρ,θ)-pseudounivex at S, or∑p i=1 ui[Gi(S)ρi +Fi(S)ρi] + ρ > 0;

(c) the assumptions specified in part (c) of Theorem 3.1 are satisfied for the feasible solu- tion (S, u, v) of (DI), and the function

R−→ p∑ i=1

q∑ j=1

v jHj(R) (3.24)

is strictly (,b, φ,0,θ)-pseudounivex at S, and φ(a) > 0⇒ a > 0. Then S= S∗, that is, S is an efficient solution of (P).

Proof. (a) Suppose to the contrary that S = S∗. Proceeding as in the proof of part (a) of Theorem 5.1, we arrive at the strict inequality

p∑ i=1

ui [ Gi(S)Fi(S∗)−Fi(S)Gi

]= 0, (3.25)

in contradiction to (3.23). Hence we conclude that S= S∗. (b) and (c) The proofs are similar to that of part (a).

1118 Multiobjective fractional subset programming

4. Duality model II

In this section, we consider a slightly different version of (DI) that allows for a greater variety of generalized (,b,φ,ρ,θ)-univexity conditions under which duality can be es- tablished. This duality model has the form

(DII)

T ∈An, u∈U0, v ∈Rq +, (4.4)

where (S,T ;·) : Ln1(X ,A,µ) → R is a sublinear function, and U0 = {u ∈ Rq : u 0,∑p i=1ui = 1}. We next show that (DII) is a dual problem for (P) by establishing weak and strong

duality theorems. As demonstrated below, this can be accomplished under numerous sets of generalized (,b,φ,ρ,θ)-univexity conditions. Here we use the functions fi(·,S), i∈ p, f (·,S,u), and h(·,v) :An→R, which are defined, for fixed S, u, and v, as follows:

fi(T ,S,u)=Gi(S)Fi(T)−Fi(S)Gi(T), i∈ p,

f (T ,S,u)= p∑ i=1

ui [ Gi(S)Fi(T)−Fi(S)Gi(T)

vjHj(T).

(4.5)

For given u∗ ∈ U0 and v∗ ∈ Rq +, let I+(u∗)= {i∈ p : u∗i > 0} and J+(v∗)= { j ∈ q : v∗j >

0}. Theorem 4.1 (weak duality). Let S and (T ,u,v), with u > 0, be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following six sets of hypotheses is satisfied:

(a) (i) f (·,T ,u) is (,b, φ, ρ,θ)-pseudounivex at T , and φ(a) 0⇒ a 0; (ii) for each j ∈ J+ ≡ J+(v), Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing,

and φ j(0)= 0; (iii) ρ+

∑ j∈J+ vj ρ j 0;

(b) (i) f (·,T ,u) is (,b, φ, ρ,θ)-pseudounivex at T , and φ(a) 0⇒ a 0; (ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0;

(iii) ρ+ ρ 0;

G. J. Zalmai 1119

(c) (i) f (·,T ,u) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each j ∈ J+, Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing, and

φ j(0)= 0; (iii) ρ+

∑ j∈J+ vj ρ j > 0;

(d) (i) f (·,T ,u) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0;

(iii) ρ+ ρ > 0;

(e) (i) f (·,T ,u) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each j ∈ J+, Hj is strictly (,b, φ j , ρ j ,θ)-pseudounivex at T , φ j is increasing,

and φ j(0)= 0; (iii) ρ+

∑ j∈J+ vj ρ j 0;

(f) (i) f (·,T ,u) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0;

(ii) h(·,v) is strictly (, b, φ, ρ,θ)-pseudounivex at T , φ is increasing, and φ(0)= 0; (iii) ρ+ ρ 0.

Then (S) ψ(T ,u,v), where ψ = (ψ1, . . . ,ψp) is the objective function of (DII).

Proof. (a) From the primal feasibility of S and (4.3) it is clear that for each j ∈ J+,Hj(S) Hj(T) and so using the properties of φ j , we obtain φ j(Hj(S)−Hj(T)) 0, which by virtue of (ii) implies that for each j ∈ J+,

( S,T ;b(S,T)DHj(T)

) −ρ jd2(θ(S,T)

) . (4.6)

(4.8)

where the second inequality follows from (iii). In view of (i), (4.8) implies that φ( f (S,T , u)− f (T ,T ,u)) 0, which because of the properties of φ, reduces to f (S,T ,u)− f (T ,T , u) 0. But f (T ,T ,u) = 0 and hence f (S,T ,u) 0, which is precisely (3.15). Therefore, the rest of the proof is identical to that of part (a) of Theorem 3.1.

(b)–(f) The proofs are similar to that of part (a).

Theorem 4.2 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following six sets of hypotheses is satisfied:

(a) (i) for each i∈ I+ ≡ I+(u), fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, and φi(0)= 0;

1120 Multiobjective fractional subset programming

(ii) for each j ∈ J+ ≡ J+(v), Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing,

and φ j(0)= 0; (iii) ρ +

∑ j∈J+ vj ρ j 0, where ρ =∑i∈I+ uiρi;

(b) (i) for each i∈ I+, fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increas- ing, and φi(0)= 0;

(ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0; (iii) ρ + ρ 0;

(c) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) for each j ∈ J+, Hj is strictly (,b, φ j , ρ j ,θ)-pseudounivex at T , φ j is increasing,

and φ j(0)= 0; (iii) ρ +

∑ j∈J+ vj ρ j 0;

(d) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) h(·,v) is strictly (,b, φ, ρ,θ)-pseudounivex at T , φ is increasing, and φ(0)= 0; (iii) ρ + ρ 0;

(e) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) for each j ∈ J+, Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing, and

φ j(0)= 0; (iii) ρ +

∑ j∈J+ vj ρ j > 0;

(f) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0; (iii) ρ + ρ > 0;

Then (S) ψ(T ,u,v).

Proof. (a) Suppose to the contrary that (S) ψ(T ,u,v). This implies that for each i ∈ p, Fi(S)/Gi(S) Fi(T)/Gi(T), and so Gi(T)Fi(S)− Fi(T)Gi(S) 0, with strict inequality holding for at least one index ∈ p. Hence for each i∈ p,

Gi(T)Fi(S)−Fi(T)Gi(S) 0=Gi(T)Fi(T)−Fi(T)Gi(T), (4.9)

which in view of the properties of φi can be expressed as φi( fi(S,T)− fi(T ,T)) 0. By virtue of (i), these inequalities imply that for each i∈ I+,

( S,T ;b

]) <−ρid2(θ(S,T)

) . (4.10)

) . (4.12)

But this contradicts (4.7), which is valid for the present case because of our hypotheses set forth in (ii). Hence (S) ψ(T ,u,v).

(b)–(f) The proofs are similar to that of part (a).

The next theorem may be viewed as a variant of Theorem 4.2; its proof is similar to that of Theorem 4.2 and hence omitted.

Theorem 4.3 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following four sets of hypotheses is satis- fied:

(a) (i) for each i∈ I1+ = ∅, fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i∈ I2+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , and for each i∈ I+ ≡ I+(u), φi is increasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

(ii) for each j ∈ J+ ≡ J+(v), Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing

and φ j(0)= 0; (iii) ρ +

∑ j∈J+ vj ρ j 0, where ρ =∑i∈I+ uiρi;

(b) (i) for each i∈ I1+ = ∅, fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i ∈ I2+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , and for each i ∈ I+, φi is in- creasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

(ii) h(·,v) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, and φ(0)= 0; (iii) ρ + ρ 0;

(c) (i) for each i ∈ I+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , φi is increasing, and φi(0)= 0;

(ii) for each j ∈ J1+ = ∅,Hj is strictly (,b, φ j , ρ j ,θ)-pseudounivex at T , for each j ∈ J2+, Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , and for each j ∈ J+, φ j is increasing

and φ j(0)= 0, where {J1+, J2+} is a partition of J+; (iii) ρ +

∑ j∈J+ vj ρ j 0;

(d) (i) for each i∈ I1+, fi(·,T) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i∈ I2+, fi(·,T) is (,b, φi, ρi,θ)-quasiunivex at T , and for each i∈ I+, φi is increasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

(ii) for each j ∈ J1+, Hj is strictly (,b, φ j , ρ j ,θ)-pseudounivex at T , for each j ∈ J2+,

Hj is (,b, φ j , ρ j ,θ)-quasiunivex at T , and for each j ∈ J+, φ j is increasing and

φ j(0)= 0, where {J1+, J2+} is a partition of J+; (iii) ρ +

∑ j∈J+ vj ρ j 0;

(iv) I1+ = ∅, J1+ = ∅, or ρ + ∑

j∈J+ vj ρ j > 0. Then (x) ψ(T ,u,v).

Theorem 4.4 (strong duality). Let S∗ be a regular efficient solution of (P), let (S,S∗; DF(S∗))=∑n

k=1DkF(S∗),χSk − χS∗k for any differentiable function F :An→R and S∈An,

1122 Multiobjective fractional subset programming

and assume that any one of the sixteen sets of hypotheses specified in Theorems 4.1, 4.2, and 4.3 holds for all feasible solutions of (DII). Then there exist u∗ ∈U and v∗ ∈Rq

+ such that (S∗,u∗,v∗) is an efficient solution of (DII) and φ(S∗)= ψ(S∗,u∗,v∗).

Proof. By Theorem 2.9, there exist u∗ ∈ U and v∗ ∈ Rq + such that (S∗,u∗,v∗) is a fea-

sible solution of (DII) and (S∗)= ψ(S∗,u∗,v∗). That (S∗,u∗,v∗) is efficient for (DII) follows from (the corresponding part of) Theorems 4.1, 4.2, and 4.3.

The proofs of the next two theorems are similar to that of Theorem 3.3.

Theorem 4.5 (strict converse duality). Let S∗ and (S,S∗;·) be as in Theorem 4.4, let (S, u, v) be a feasible solution of (DII) such that f (S∗, S, u) 0, and assume that either one of the two sets of hypotheses specified in parts (a) and (b) of Theorem 4.1 is satisfied for the fea- sible solution (S, u, v) of (DII). Assume, furthermore, that f (·, S, u) is strictly (,b, φ, ρ,θ)- pseudounivex at S and that φ(a) > 0⇒ a > 0. Then S= S∗, that is, S is an efficient solution of (P).

Theorem 4.6 (strict converse duality). Let S∗ and (S,S∗;·) be as in Theorem 4.4, let (S, u, v) be a feasible solution of (DII) such that f (S∗, S, u) 0, and assume that any one of the four sets of hypotheses specified in parts (c)–(f) of Theorem 4.1 is satisfied for the feasible solution (S, u, v) of (DII). Assume, furthermore, that f (·, S, u) is (,b, φ, ρ,θ)-quasiunivex at S and that φ(a) > 0⇒ a > 0. Then S= S∗, that is, S is an efficient solution of (P).

5. Duality model III

In this section, we formulate a more general duality model for (P) with a more flexible structure which will allow us to establish duality under various generalized (,b,φ,ρ,θ)- univexity hypotheses that can be imposed on certain combinations of the problem func- tions. This will be accomplished by utilizing a partitioning scheme which was originally proposed in [2] for the purpose of constructing generalized dual problems for nonlin- ear programs with point-functions. Prior to formulating this duality model, we need to introduce some additional notation.

Let {J0, J1, . . . , Jm} be a partition of the index set q; thus Jr ⊂ q for each r ∈ {0,1, . . . ,m}, Jr ∩ Js = ∅ for each r,s ∈ {0,1, . . . ,m} with r = s, and ∪m

r=0Jr = q. In addition, we will make use of the real-valued functions Πi(·,T ,v), Π(·,T ,u,v), and Λt(·,v) defined, for fixed T , u, and v, on An by

Πi(R,T ,v)=Gi(T)

ui

{ Gi(T)

[ Fi(R) +

(5.1)

Maximize (F1(T) +

T ∈An, u∈U0, v ∈Rq +, (5.5)

where (S,T ;·) : Ln1(X ,A,µ)→R is a sublinear function. We next show that (DIII) is a dual problem for (P) by proving weak and strong duality

theorems.

Theorem 5.1 (weak duality). Let S and (T ,u,v), with u > 0, be arbitrary feasible solu- tions of (P) and (DIII), respectively, and assume that any one of the following four sets of hypotheses is satisfied:

(a) (i) Π(·,T ,u,v) is (,b, φ, ρ,θ) -pseudounivex at T , and φ(a) 0⇒ a 0; (ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and

φt(0)= 0; (iii) ρ+

∑m t=1 ρt 0;

(b) (i) Π(·,T ,u,v) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each t ∈m, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , φt is increas-

ing, and φt(0)= 0; (iii) ρ+

∑m t=1 ρt 0;

(c) (i) Π(·,T ,u,v) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and

φt(0)= 0; (iii) ρ+

∑m t=1 ρt > 0;

(d) (i) Π(·,T ,u,v) is prestrictly (,b, φ, ρ,θ)-quasiunivex at T , and φ(a) 0⇒ a 0; (ii) for each t ∈m1, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and

φt(0)= 0, for each t ∈m2 = ∅, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at

T , φt is increasing, and φt(0)= 0, where {m1, m2} is a partition of m; (iii) ρ+

∑m t=1 ρt 0.

Then (S) ω(T ,u,v), where ω = (ω1, . . . ,ωp) is the objective function of (DIII).

1124 Multiobjective fractional subset programming

) 0.

(5.6)

Since S∈ F and v 0, it is clear from (5.4) that for each t ∈m,

Λt(S,v)= ∑ j∈Jt

v jHj(T)=Λt(T ,v), (5.7)

and so using the properties of φt, we get φt ( Λt(S,v)−Λt(T ,v)

) 0, which in view of (ii)

implies that for each t ∈m,

) −ρtd2(θ(S,T)

) . (5.8)

(5.10)

where the second inequality follows from (iii). Because of (i), this inequality implies that φ(Π(S,T ,u,v)−Π(T ,T ,u,v)) 0. But φ(a) ≥ 0 ⇒ a ≥ 0, and so we get Π(S,T ,u,v) Π(T ,T ,u,v) = 0. Inasmuch as S ∈ F, u > 0, v 0, and Gi(T) > 0, i ∈ p, this inequality yields

p∑ i=1

] Gi(S)

} 0. (5.11)

Since u > 0, this inequality implies that( G1(T)F1(S)− [F1(T) +Λ0(T ,v)

] G1(S), . . . ,Gp(T)Fp(S)

(S)= ( F1(S) G1(S)

(b)–(d) The proofs are similar to that of part (a).

Theorem 5.2 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DIII), respectively, and assume that any one of the following six sets of hypotheses is satis- fied:

(a) (i) for each i∈ I+ ≡ I+(u), Πi(·,T ,v, ) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, and φi(0)= 0;

(ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt 0;

(b) (i) for each i ∈ I+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , φi is in- creasing, and φi(0)= 0;

(ii) for each t ∈m, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , φt is increas- ing, and φt(0)= 0;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt 0;

(c) (i) for each i ∈ I+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , φi is in- creasing, and φi(0)= 0;

(ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt > 0;

(d) (i) for each i ∈ I1+ = ∅, Πi(·,T ,v) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i∈ I2+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , and for each i∈ I+, φi is increasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

(ii) for each t ∈m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt 0;

(e) (i) for each i ∈ I+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , φi is in- creasing, and φi(0)= 0;

(ii) for each t ∈m1 = ∅, Λt(·,v) is strictly (,b, φt , ρt,θ)-pseudounivex at T , φt is

increasing, and φt(0) = 0, and for each t ∈m2, Λt(·,v) is (,b, φt, ρt,θ)-quasi-

univex at T , and for each t ∈m, φt is increasing and φt(0)= 0, where {m1, m2} is a partition of m;

(iii) ∑

i∈I+ uiρi + ∑m

t=1 ρt 0;

(f) (i) for each i ∈ I1+, Πi(·,T ,v) is strictly (,b, φi, ρi,θ)-pseudounivex at T , for each i∈ I2+, Πi(·,T ,v) is prestrictly (,b, φi, ρi,θ)-quasiunivex at T , and for each i∈ I+, φi is increasing and φi(0)= 0, where {I1+,I2+} is a partition of I+;

1126 Multiobjective fractional subset programming

(ii) for each t ∈ m1, Λt(·,v) is strictly (,b, φt , ρt,θ)-pseudounivex at T , for each

t ∈m2, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , and for each t ∈m, φt is in-

creasing and φt(0)= 0, where {m1,m2} is a partition of m; (iii)

∑ i∈I+ uiρi +

(iv) I1+ = ∅, m1 = ∅, ∑

t=1 ρt > 0. Then (S) ω(T ,u,v).

Proof. (a) Suppose to the contrary that (S) ω(T ,u,v). This implies that for each i∈ p, Gi(T)Fi(S)− [Fi(T) +Λ0(T ,v)]Gi(S) 0, with strict inequality holding for at least one index ∈ p. Using these inequalities along with the primal feasibility of S and nonnega- tivity of v, we see that

Πi(S,T ,v)=Gi(T)

=Πi(T ,T ,v).

(5.14)

(5.15)

) , (5.17)

where the second inequality follows from (iii). Obviously, this inequality contradicts (5.9), which is valid for the present case because of (ii). Hence we must have (S) ω(T ,u,v).

(b)–(f) The proofs are similar to that of part (a).

G. J. Zalmai 1127

Theorem 5.3 (strong duality). Let S∗ be a regular efficient solution of (P), let (S,S∗; DF(S∗))=∑n

k=1DkF(S∗),χSk − χS∗k for any differentiable function F :An→R and S∈An, and assume that any one of the ten sets of hypotheses specified in Theorems 5.1 and 5.2 holds for all feasible solutions of (DIII). Then there exist u∗ ∈ U and v∗ ∈ Rq

+ such that (S∗,u∗,v∗) is an efficient solution of (DIII) and (S∗)= ω(S∗,u∗,v∗).

Proof. By Theorem 2.9, there exist u∗ ∈U and v ∈Rq + such that

⟨ p∑ i=1

] +

(5.18)

v jHj(S∗)= 0, j ∈ q. (5.19)

Now if we let v∗j = v j /Gi(S∗) for each j ∈ J0, v∗j = v j for each j ∈ q \ J0, and observe that Λ0(S∗,v∗)= 0, then (5.18) and (5.19) can be rewritten as follows:

⟨ p∑ i=1

0 ∀Sk ∈A, k ∈ n,

(5.20)∑ j∈Jt

v∗j Hj(S∗)= 0, t ∈m. (5.21)

From (5.20) and (5.21) it is clear that (S∗,u∗,v∗) is a feasible solution of (DIII). Proceed- ing as in the proof of Theorem 3.3, it can easily be verified that it is an efficient solution of (DIII).

We next show that certain modifications in Theorem 5.1 lead to a number of strict converse duality results for (P) and (DIII).

Theorem 5.4 (strict converse duality). Let S∗ and be as in Theorem 5.3, let (S, u, v) be a feasible solution of (DIII) such that

p∑ i=1

] Gi(S∗)

} 0, (5.22)

and assume that any one of the four sets of hypotheses specified in Theorem 5.1 is satisfied for the feasible solution (S, u, v) of (DIII). Assume furthermore that any one of the following corresponding conditions holds:

(a) Π(·, S, u, v) is strictly (,b, φ, ρ,θ)-pseudounivex at S, and φ(a) > 0⇒ a > 0; (b) Π(·, S, u, v) is (,b, φ, ρ,θ)-quasiunivex at S, and φ(a) > 0⇒ a > 0;

1128 Multiobjective fractional subset programming

(c) Π(·, S, u, v) is (,b, φ, ρ,θ)-quasiunivex at S, and φ(a) > 0⇒ a > 0; (d) Π(·, S, u, v) is (,b, φ, ρ,θ)-quasiunivex at S, and φ(a) > 0⇒ a > 0.

Then S= S∗.

Proof. The proof is similar to that of Theorem 3.3.

Evidently, (DIII) contains a number of important special cases which can easily be identified by appropriate choices of the partitioning sets J0, J1, . . . , Jm, and the sublinear function (S,T ;·). We conclude this section by briefly looking at a few of these special cases. In each case, we specify the required conditions for duality by specializing part (a) of Theorem 5.2.

If we let J0 = q, then (DIII) takes the following form: (DIIIa)

Maximize (F1(T) +

(5.24)

(DIIIa) is dual to (P) if for each i∈ I+, the function

R−→Gi(T)

] Gi(R) (5.25)

is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0)= 0, and ∑

i∈I+ uiρi 0. If we let J1 = q, then (DIII) becomes

(DIIIb)

G. J. Zalmai 1129

(DIIIb) is dual to (P) if for each i ∈ I+, the function R→ Gi(T)Fi(R)− Fi(T)Gi(R) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0) = 0, the function R→∑q

j=1 vjHj(R) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, φ(0)= 0, and ∑

i∈I+ uiρi + ρ 0.

If we choose J0 =∅, then we obtain the following special case of (DIII): (DIIIc)

Maximize ( F1(T) G1(T)

v jHj(T) 0, t ∈m. (5.29)

(DIIIc) is dual to (P) if for each i ∈ I+, the function R→ Gi(T)Fi(R)− Fi(T)Gi(R) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0) = 0, for each t ∈ m, Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex at T , φt is increasing, φt(0) = 0, and

∑ i∈I+ uiρi +∑m

t=1 ρt 0. If we set m= q, Jt = {t}, t ∈ q, then (DIII) reduces to the following problem:

(DIIId)

vjHj(T) 0, j ∈ q. (5.31)

(DIIId) is dual to (P) if for each i∈ I+, the function R→ Gi(T)Fi(R)− Fi(T)Gi(R) is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0)= 0, for each j ∈ q, R→ vjHj(R) is (,b, φ j , ρ j ,θ)-quasiunivex at T , φ j is increasing, φ j(0) = 0, and

∑ i∈I+ uiρi +∑q

j=1 ρ j 0. If we let Jt =∅, t = 2,3, . . . ,m, then we get the following special case of (DIII):

(DIIIe)

subject to (5.5) and

vjHj(T) 0. (5.34)

(DIIIe) is dual to (P) if for each i∈ I+, the function

R−→Gi(T)

] Gi(R) (5.35)

is strictly (,b, φi, ρi,θ)-pseudounivex at T , φi is increasing, φi(0)= 0, the function R→∑ j∈J1 vjHj(R) is (,b, φ, ρ,θ)-quasiunivex at T , φ is increasing, φ(0)= 0, and

∑ i∈I+ uiρi +

ρ 0. In a similar manner, one can obtain a vast number of duality theorems for (P) by

specializing the other nine sets of conditions for (DIIIa)–(DIIIe) and other special cases of (DIII).

6. Duality model IV

In this section, we present another general duality model for (P) that is different from (DIII) in that here in constructing the constraints we not only use a partition of the index set q, but also a partition of the set p. A parametric point-function version of this dual problem was considered earlier in [4].

Let {I0,I1, . . . ,Ik} be a partition of p such that K ≡ {0,1, . . . ,k} ⊂M ≡ {0,1, . . . ,m}, and let the function t(·,S,u,v) :An→R be defined, for fixed S, u, and v, by

t(T ,S,u,v)= ∑ i∈It

Maximize ( F1(T) G1(T)

T ∈An, u∈U , v ∈Rq +, (6.5)

where (S,T ;·) : Ln1(X ,A,µ)→R is a sublinear function. The next two theorems show that (DIV) is a dual problem for (P).

Theorem 6.1 (weak duality). Let S and (T ,u,v) be arbitrary feasible solutions of (P) and (DIV), respectively, and assume that any one of the following six sets of hypotheses is satis- fied:

(a) (i) for each t ∈ K , t(·,T ,u,v) is strictly (,b,φt,ρt,θ)-pseudounivex at T , φt is increasing, and φt(0)= 0;

(ii) for each t ∈M \K , Λt(·,v) is (,b,φt,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

t∈M ρt 0;

(b) (i) for each t ∈ K , t(·,T ,u,v) is prestrictly (,b,φt ,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(ii) for each t ∈M \K , Λt(·,v) is strictly (,b,φt,ρt,θ)-pseudounivex at T , φt is in- creasing, and φt(0)= 0;

(iii) ∑

t∈M ρt 0;

(c) (i) for each t ∈ K , t(·,T ,u,v) is prestrictly (,b,φt ,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(ii) for each t ∈M \K , Λt(·,v) is (,b,φt,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

t∈M ρt > 0;

(d) (i) for each t ∈ K1 = ∅, t(·,T ,u,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , for each t ∈ K2, t(·,T ,u,v) is prestrictly (,b, φt , ρt,θ)-quasiunivex at T , and for each t ∈ K , φt is increasing and φt(0)= 0, where {K1,K2} is a partition of K ;

(ii) for each t ∈M \K , Λt(·,v) is (,b,φt,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(iii) ∑

t∈M ρt 0;

(e) (i) for each t ∈ K , t(·,T ,u,v) is prestrictly (,b,φt ,ρt,θ)-quasiunivex at T , φt is increasing, and φt(0)= 0;

(ii) for each t ∈ (M \K)1 = ∅, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , for each t ∈ (M \K)2,Λt(·,v) is (,b, φt, ρt,θ)-quasiunivex atT , and for each t ∈ M \K , φt is increasing and φt(0)= 0, where {(M \K)1, (M \K)2} is a partition of M \K ;

(iii) ∑

1132 Multiobjective fractional subset programming

(f) (i) for each t ∈ K1, t(·,T ,u,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , for each t ∈ K2, t(·,T ,u,v) is prestrictly (,b, φt, ρt,θ)-quasiunivex at T , and for each t ∈ K , φt is increasing and φt(0)= 0, where {K1,K2} is a partition of K ;

(ii) for each t ∈ (M \K)1, Λt(·,v) is strictly (,b, φt, ρt,θ)-pseudounivex at T , φt is increasing, and φt(0) = 0, and for each t ∈ (M \K)2, Λt(·,v) is (,b, φt, ρt,θ)- quasiunivex at T , and for each t ∈M \K , φt is increasing and φt(0) = 0, where {(M \K)1,(M \K)2} is a partition of M \K ;

(iii) ∑

∑ t∈M ρt > 0.

Then (S) δ(T ,u,v), where δ = (δ1, . . . ,δp) is the objective function of (DIV).

(6.6)

(6.7)

) −ρtd2(θ(S,T)

) . (6.8)

G. J. Zalmai 1133

) 0,

(6.10)

which contradicts (6.3). Therefore, (S) δ(T ,u,v). (b)–(f) The proofs are similar to that of part (a).

Theorem 6.2 (strong duality). Let S∗ be a regular efficient solution of (P), let (S,S∗; DF(S∗)) =∑n

k=1DkF(S∗),χSk − χS∗k for any differentiable function F : An → R and S ∈ An, and assume that any one of the six sets of hypotheses specified in Theorem 6.1 holds for all feasible solutions of (DIV). Then there exist u∗ ∈U and v∗ ∈Rq

+ such that (S∗,u∗,v∗) is an efficient solution of (DIV) and (S∗)= δ(S∗,u∗,v∗).

Proof. The proof is similar to that of Theorem 4.4.

Various special cases of (DIV) can easily be generated by appropriate choices of the partitioning sets I0,I1, . . . ,Ik, J0, and (S,T ;·).

References

[1] H. W. Corley, Optimization theory for n-set functions, J. Math. Anal. Appl. 127 (1987), no. 1, 193–205.

[2] B. Mond and T. Weir, Generalized concavity and duality, Generalized Concavity in Optimization and Economics (S. Schaible and W. T. Ziemba, eds.), Academic Press, New York, 1981, pp. 263–279.

[3] R. J. T. Morris, Optimal constrained selection of a measurable subset, J. Math. Anal. Appl. 70 (1979), no. 2, 546–562.

[4] X. M. Yang, Generalized convex duality for multiobjective fractional programs, Opsearch 31 (1994), no. 2, 155–163.

[5] G. J. Zalmai, Efficiency conditions and duality models for multiobjective fractional subset pro- gramming problems with generalized (,α,ρ,θ)-V-convex functions, Comput. Math. Appl. 43 (2002), no. 12, 1489–1520.

[6] , Generalized (,b,φ,ρ,θ)-univex n-set functions and global semiparametric sufficient efficiency conditions in multiobjective fractional subset programming, to appear in Int. J. Math. Math. Sci.

G. J. Zalmai: Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI 49855, USA

E-mail address: [email protected]

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